Question

**Given:** Class intervals $4$–$8$, $8$–$12$, $12$–$16$, $16$–$20$ with frequencies $3,\ \lambda,\ 4,\ 7$ and variance $\sigma^2 = 19$. **Step 1: Find midpoints.** $$x_i = 6,\ 10,\ 14,\ 18$$ **Step 2: Total frequency.** $$N = 3 + \lambda + 4 + 7 = 14 + \lambda$$ **Step 3: Compute the mean.** $$\sum f_i x_i = 3(6) + \lambda(10) + 4(14) + 7(18) = 18 + 10\lambda + 56 + 126 = 200 + 10\lambda$$ $$\mu = \frac{200 + 10\lambda}{14 + \lambda} = 10 + \frac{60}{14 + \lambda}$$ For $\mu$ to be a rational value, $(14 + \lambda)$ must be a factor of $60$. Possible values: $\lambda = 1\ (\mu = 14),\ \lambda = 6\ (\mu = 13),\ \lambda = 16\ (\mu = 12)$. **Step 4: Apply variance condition.** $$\sum f_i x_i^2 = 3(36) + \lambda(100) + 4(196) + 7(324) = 108 + 100\lambda + 784 + 2268 = 3160 + 100\lambda$$ $$\sigma^2 = \frac{\sum f_i x_i^2}{N} - \mu^2 = 19$$ **For $\lambda = 6,\ \mu = 13$:** $$\sigma^2 = \frac{3160 + 600}{20} - 169 = \frac{3760}{20} - 169 = 188 - 169 = 19 \checkmark$$ **Step 5: Compute answer.** $$\lambda + \mu = 6 + 13 = 19$$ **Answer: (D)**